| Directions: (31 - 35) |
| If \[y=f\left( u \right)\]is a differentiable function of u and \[u=g\left( x \right)\]is a differentiable function of x, then \[y=f\left[ \left( g\left( x \right) \right. \right]\]is a differentiable function of x and\[\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\]. This rule is also known as CHAIN RULE. |
| Based on the above information, find the derivative of functions w.r.t. x in the following questions. |
A) \[\frac{-\sin \sqrt{x}}{2\sqrt{x}}\]
B) \[\frac{\sin \sqrt{x}}{-\sin \sqrt{x}}\]
C) \[sin\sqrt{x}\]
D) \[-\sin \sqrt{x}\]
Correct Answer: A
Solution :
Let \[y=\cos \sqrt{x}\] \[\therefore \,\,\frac{dy}{dx}=\frac{dy}{dx}\left( \cos \,\sqrt{x} \right)=-\sin \sqrt{x}\,.\,\frac{d}{dx}\left( \sqrt{x} \right)\] \[=-\sin \sqrt{x}\times \frac{1}{2\sqrt{x}}=\frac{-\sin \sqrt{x}}{2\sqrt{x}}\]
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